Method for correcting the effects of interdetector band broadening

ABSTRACT

Chromatographic separations are often characterized by multiple detectors through which the sample flows serially. As the sample flows between detectors, it becomes progressively diluted due to mixing and diffusion. This phenomenon is traditionally called interdetector “band broadening” and often results in significant distortion of the sample&#39;s derived physical properties such as molar mass. A method to characterize the broadening present in a chromatographic system, and an algorithm whereby the narrow peaks of the upstream detector are numerically broadened so they can be compared to the broadened peaks of the downstream detector, is described. Although the technique results in some loss of resolution, its stability and generality allow it a broad range of application. Examples are presented for data collected by dRI, MALS, UV, and viscometric detectors.

RELATED PATENTS

The methods disclosed in this specification will have particularrelevance to a variety of patents and applications by the assignee andinventor. They include the following:

Steven P. Trainoff and Philip J. Wyatt, U.S. Pat. No. 6,651,009, Issued18 Nov. 2003, Method for determining average solution properties ofmacromolecules by the injection method.

Philip J. Wyatt, U.S. Pat. No. 6,411,383 25, issued 25 Jun. 2002,“Method for measuring the 2^(nd) virial coefficient.”

Steven P. Trainoff, U.S. Pat. No. 6,180,906, issued 30 Jan. 2001,“Electrode design for electrical field flow fractionation.”

Steven P. Trainoff, et al., U.S. Pat. No. 6,128,080, issued 3 Oct. 2000,“Extended range interferometric refractometer.”

Gary R Janik, et al., U.S. Pat. No. 5,900,152, issued 4 May 1999,“Apparatus to reduce inhomogeneities in optical flow cells.”

Gary R Janik, et al., U.S. Pat. No. 5.676,830, issued 14 Oct. 1997,“Method and apparatus for reducing band broadening in chromatographicdetectors.”

Philip J. Wyatt, et al., U.S. Pat. No. 5,530,540, issued 25 Jun. 1996,“Light scattering measurement cell for very small volumes.”

David W. Shortt, U.S. Pat. No. 5,528,366, issued 18 Jun. 1996,“Precision determination for molecular weights.”

Philip J. Wyatt, U.S. Pat. No. 5,305,071, issued 19 Apr. 1994,“Differential refractometer.”

BACKGROUND

The analysis of macromolecular species in solution by liquidchromatographic techniques is usually achieved by preparing a sample inan appropriate solvent and then injecting an aliquot thereof into achromatograph. The chromatograph includes fractionation devices ofvarious types that separate the sample as it passes through them. Onceseparated by such means, generally based on size, mass, or columnaffinity, the samples are subjected to analysis by means of lightscattering, refractive index, UV, viscometric response, etc. Forexample, in order to determine the mass and size distributions of aparticular sample whose separation is performed by size exclusioncolumns, the chromatograph would measure the sample sequentially bymultiangle light scattering, MALS followed by a differentialrefractometer, dRI. The dRI determines the concentration while the MALSunit measures the excess Rayleigh ratio as a function of angle of eacheluting fraction. For molecules very much smaller than the wavelength ofthe incident light, light scattering measurements at a single angle mayoften be sufficient.

As each fraction passes through a detection device, it produces a signaloften referred to as a “band” or “peak.” Because of dispersion andmixing effects, these bands are broadened somewhat each time the samplepasses through a different device. Consider a sample comprised of a lowconcentration aliquot of a monodisperse protein. In this event, theRayleigh excess ratio is directly proportional to the molar mass and theconcentration. The light scattering signal and the concentration signalshould be of identical shape and would overlay perfectly were the tworesponses normalized to have the same areas. However, as the samplepasses from the MALS detector and enters the dRI detector, it passesthrough intermediate regions and connections that contribute to thedispersion and mixing of the sample. Many sources of band broadening andsome of their remedies are discussed in detail by Yau, et al. in theirbook “Modem size exclusion liquid chromatography” published by JohnWiley & Sons in 1979. In the above example, the dRI signal will alwaysappear somewhat broadened with respect to the MALS signals.

The effect of broadening is shown clearly in FIG. 1 which shows achromatogram of a sample of bovine serum albumen, BSA, in an aqueousbuffer along with the computed molar mass, uncorrected for the effectsof broadening. The uncorrected molar mass is shown in trace 1. It ispresented divided by the molar mass of the monomer, M_(w0), so that thevertical axis for the monomer reads 1, the dimer reads 2, etc. Trace 2corresponds to the 90° light scattering signal as a function of time.Trace 3 corresponds to the dRI signal. The aggregation states have beenfractionated so that the rightmost peak region, from 16 minutes to 18minutes, is data from the pure monomer, from 14 minutes to 16 minutes isfrom the dimer, etc. In each peak, broadening causes the concentrationat the center of the peak to be suppressed and that in the “wings” to beenhanced. Combining this broadened dRI signal with the MALS signals todetermine molar mass at each elution time causes the molar massdetermined near the peak's center to be systematically overestimated,while that in the wings to be underestimated. The situation reverseswhen the concentration detector is upstream of the MALS detector. Thisgives rise to the non-constant molar mass data within each peak region.If there were no band broadening, the data would consist of a series ofplateaus at 1 for the monomer, 2 for the dimer, etc. As the fractionatedsample becomes more polydisperse, the effects of band broadening becomeless apparent (but are still present). For a well fractionated samplecomprised of discrete aggregate states, the problem is obvious. In FIG.1, the monomer peak is well resolved so the broadening problem isclearest. The aggregates are progressively less well resolved so theyshow the problem to a lesser extent.

In order to compensate for these distortions, a variety of methods havebeen employed. Most are based upon the work of L. H. Tung presented inhis 1969 paper published in the Journal of Applied Polymer Science,volume 13, p 775 et seq. The afore-referenced book by Yau et al.discusses some of the means to correct band broadening. Throughout thechromatography literature, many papers address means to perform thesecorrections, i.e. to restore the broadened band to the shape it wouldhave had were no broadening sources present. Most of the techniques arenumerically unstable and can result in physically unreasonable results,such as negative concentrations, or negative scattered intensities, andtherefore are rarely used. The characterization of proteins by MALSmeasurements represents an area of particular importance, yet even here,band-broadening corrections are rarely seen because of the difficultiesassociated with their implementation.

Band broadening effects occur in a variety of multi-detectorimplementations of liquid chromatography. For example, if a sample is tobe measured by an on-line viscometer so that its intrinsic viscosity maybe determined, then a dRI will also be needed. As the sample passes fromthe dRI to the viscometer, it will experience band broadening resultingin a spreading of the specific viscosity curve with respect to theconcentration curve derived from the dRI measurement. Again, thecomputational difficulties, associated with restoring the broadenedsignal to conform to what it would have been without broadening, oftenresult in questionable results, especially if the samples are indeedmonodisperse such as unaggregated proteins.

It is the purpose of the present invention to provide an analyticalmethod by which band-broadening effects may be corrected in softwarewith both ease and precision. Another objective of the invention is toprovide a new approach to such corrections whose departure from theconventional approach is sharply distinguished therefrom. The presentinvention is expected to be of greatest utility for the analysis ofseparations of protein samples. A consequence of this application willbe the more precise characterization of protein conjugates as well asaggregate states. A further application of this invention is its abilityto improve the determination of intrinsic viscosity measurements bycorrecting the band broadening effects associated with the samplepassage between a concentration detector and a viscometer. An importantapplication of the invention will be its application to the rapiddetermination of the 2^(nd) virial coefficient of an unfractionatedsample as discussed, for example, in U.S. Pat. No. 6,411,383 and theapplication Ser. No. 10/205,637 filed 24 Jul. 2002, now in process ofissuance, by Trainoff and Wyatt. The successful implementation of thatmethod depends critically upon the ability to determine accurately thesquare of the concentration at each elution volume. This requires anaccurate measurement of the concentration at each such elution volume.The latter determinations are made generally on unfractionated sampleswhose detector responses are a single peak whose mass composition is thesame at each interval collected. This single peak is broadened as itproceeds through the series of detectors leading to systematic errors inthe derived results, unless the effect of broadening is corrected.

BRIEF DESCRIPTION OF THE DRAWINGS

FIG. 1 shows an uncorrected plot of the molar mass of a BSA sample withthe dRI detector downstream of MALS detector.

FIG. 2 shows the hybrid broadening model fitted to the 90° lightscattering trace of a 5 μl injection of 200 kD polystyrene in toluene.

FIG. 3 shows the data of FIG. 1 corrected by the method of theinvention.

FIG. 4 shows an uncorrected plot of intrinsic viscosity versus elutionvolume for a BSA sample with a dRI detector downstream of viscometer.

FIG. 5 shows the data of FIG. 4 corrected by the method of theinvention. Trace 11 is the corrected intrinsic viscosity

FIG. 6 shows an uncorrected plot of the molar mass of a BSA sample witha UV detector upstream of the MALS.

FIG. 7 shows the data of FIG. 6 corrected by the method of theinvention.

FIG. 8 is a graphical representation of the algorithm.

SUMMARY OF THE INVENTION

The present invention comprises a method to correct for band broadeningeffects occurring between detectors of a liquid chromatograph. Ratherthan correct the band broadening produced at a second detector byrestoring the band to the form of the first, the method broadens theband of the first detector to conform to the broadened band at thesecond. This results in a simpler, more robust, and more easilyimplemented means to correct band-broadening effects at all detectors.Although the method can result in reduced peak resolution when peaksoverlap partially, areas of major application are expected to be formonodisperse samples with well-separated and resolved peaks. Theinvention is applicable as well to any number of serially connecteddetectors.

DETAILED DESCRIPTION OF THE INVENTION

Molecules in solution are generally characterized by their weightaveraged molar mass M_(w), their mean square radius <r _(g) ²>, and thesecond virial coefficient A₂. The latter is a measure of the interactionbetween the molecules and the solvent. For unfractionated solutions,these properties may be determined by measuring the manner by which theyscatter light using the method described by Bruno Zimm in his seminal1948 paper that appeared in the Journal of Chemical Physics, volume 16,pages 1093 through 1099. More recently, the method discussed in U.S.Pat. No. 6,651,009 by Trainoff and Wyatt, represents an advancedtechnique that will replace the more traditional Zimm approach.

The light scattered from a small volume of the solution is measured overa range of angles and concentrations. The properties derived from thelight scattering measurements are related through the formula developedby Zimm:R(θ)=K*M _(w) cP(θ)[1−2A ₂ M _(w) cP(θ)]+O(c ³),   (1)where R(θ) is the measured excess Rayleigh ratio in the direction θ perunit solid angle defined as R(θ)=[I_(s)(θ)−I_(solv)(θ)]r²/I₀V, I_(s)(θ)is the intensity of light scattered by the solution per unit solidangle, I_(solv)(θ) is the intensity of light scattered from the solventper unit solid angle, I₀ is the incident intensity, r is the distancefrom the scattering volume to the detector, V is the illuminated volumeseen by the detectors, P(θ) is the form factor of the scatteringmolecules defined as

${{P(\theta)} = {\lim\limits_{c->0}{{R(\theta)}/{R(0)}}}},$K*=4π²(dn/dc)²n₀ ²/(N_(a)λ₀ ⁴), N_(a) is Avogadro's number, dn/dc is therefractive index increment, n₀ is the solvent refractive index, c is theconcentration, and λ₀ is the wavelength of the incident light in vacuum.The incident collimated light beam is assumed vertically polarized withrespect to the plane containing the light scattering detectors. The formfactor is related to the mean square radius <r_(g) ²> by

$\begin{matrix}{{P(\theta)} = {1 - {\frac{16\pi^{2}n_{0}^{2}}{3\lambda_{0}^{2}}\left\langle r_{g}^{2} \right\rangle{\sin^{2}\left( {\theta/2} \right)}} + {O\left( {{\sin^{4}(\theta)}/2} \right)}}} & (2)\end{matrix}$

Multiangle light scattering instruments measure the amount of lightscattered from a fluid sample as a function of both angle and time. Formolecules very small compared to the wavelength of the incident light,light scattering measurements may be restricted to a single scatteringangle, such as 90°, since there would be no measurable variation of thescattered intensity with angle, however in this limit, the molar massmay be measured, but the mean square radius may not.

When the instrument is used to analyze the eluant of a separationsystem, such as size exclusion chromatography, SEC, the samplecomposition and concentration change with time. Furthermore, SEC columnstypically dilute the sample concentration so that the second and thirdterms on the right hand side of Eq. (1) can be neglected compared to thefirst. Formally this is the case when2A₂cM_(w)<<1.  (3)

Typically, for chromatographic separations combined with lightscattering instrumentation, the above condition is assumed unless A2 isknown from prior experiment, in which case such value may be used in Eq.1 directly. In addition, if the separation is performed properly, ateach point in time the sample is essentially monodisperse in size, andusually in mass. If the sample is also monodisperse in mass we have thefurther simplification thatR(θ,t)=K*Mc(t)P(θ,t).  (4)

Therefore, the light scattering signal should be directly proportionalto the concentration signal and the molar mass can be computed by theratio

$\begin{matrix}{M = {\frac{1}{K^{*}{c(t)}}{\lim\limits_{\theta->0}{{R\left( {\theta,t} \right)}.}}}} & (5)\end{matrix}$

A common test for the correct operation of a MALS system is to inject anearly monodisperse sample through a separation column, and then measureboth the light scattering and the concentration signals for amonodisperse peak. If normalized, they should overlay exactly at allelution times corresponding to the constancy expressed by Eq. (5).However, if the concentration detector is downstream of the MALSdetector, two effects will change the shape of the peak: mixing anddiffusion. When this is the case, the peak appears broadened; the centerof the peak is suppressed and the “wings” are enhanced. Therefore, whenthe ratio in Eq. (5) is formed, it will not be constant and the molarmass derived near the peak will be systematically overestimated, whilethat near the wings will be underestimated. The situation is reversedwhen the concentration detector is upstream of the MALS detector.

It should be pointed out that samples injected into a chromatographwithout fractionation, often described as flow injection, alsoexperience band broadening during their passage from one detector toanother. Although the molar mass distribution at each successive elutionvolume are expected to remain constant, band broadening effects willaffect various weight average properties derived from downstreambroadening at selected detectors. Thus an injected sample peak comprisedof a polydisperse molar mass distribution will broaden as it passes, forexample, between a light scattering detector and a dRI concentrationdetector. When the unfractionated light scattering signals are combinedwith the broadened concentration detector signal, the derived weightaveraged molar mass near the center of the peak will be calculated to belarger while the corresponding values in the wings will be smaller. Thisresult is exactly equivalent to that seen for fractionated samples.

Sample mixing primarily depends on the geometry of the system: thelength and diameter of the tubing, the number of unions, the roughnessof the surfaces, and the number of inclusions caused by mechanicaldefects, and the geometry of the measurement cells. The effect of mixingis primarily independent of the sample composition in the limit that thebulk fluid properties, such as viscosity, are not substantively changedby the sample. Diffusion, in contrast, depends explicitly on the sampleproperties.

It is instructive to estimate the relative importance of diffusion andmixing for a sample flowing between two detectors. For uniform spheresthe Stokes-Einstein relation

$\begin{matrix}{{D_{T} = \frac{k_{B}T}{6{\pi\eta}\; r}},} & (6)\end{matrix}$relates the diffusion constant D_(T) to the sphere radius r, where k_(B)is Boltzmann's constant, T is the absolute temperature, and η is thesolvent viscosity.

There are many sources of mixing, but to estimate the scale, considerthe simple model of a well-mixed reservoir. Assume that the lightscattering flow cell is a mixing chamber with a volume V. Let the flowrate be denoted f, and assume that the concentration of the sampleentering the chamber is c_(i)(t). The concentration of the sampleexiting the flow cell is given byc′(t)=[c _(i)(t)−c(t)]/t _(f,)  (7)where t_(f)=V/f is the time to fill the reservoir. Consider the effectof a narrow pulse, containing a mass m₀ of sample injected into the flowcell so that c_(i)(t)=m₀δ(t). The concentration of the sample exitingthe cell is then

$\begin{matrix}{{{c(t)} = {\frac{m_{0}}{t_{f}}{\exp\left( {{- t}/t_{f}} \right)}{\theta(t)}}},} & (8)\end{matrix}$where θ(t)=0 for t<0 and θ(t)=1 for t≧0. To put this in context, assumethat the flow cell has a volume of V=80 μl and is fed by a flow rate off=1.0 ml/min. The exponential time constant for the broadening ist_(f)=4.8 sec. If this mixing chamber is fed and drained by capillarytubing with an internal diameter of 0.25 mm, the initially localizedconcentration pulse is smeared out over a length of tubing 1.6 m inlength. Of course, when the sample flows through the light scatteringcell, it is not “well mixed,” but this is a good estimate of the lengthscale of the mixing.

Next, consider the effect of diffusion for a small molecule. Themolecules will diffuse, on average, a distance<|x|>=√{square root over (2D _(T) t)}  (9)where, D_(T) is the molecule's translational diffusion constant and t istime. Again, to put this in context, consider the protein bovine serumalbumen, BSA, which has a diffusion constant of 7.7×10⁻⁷ cm²/sec. Howfar does the concentration spike diffuse in 4.8 sec ? Equation (9)indicates that it will diffuse along the tubing a mere 27 μm !Therefore, we conclude that the dominant mixing mechanism isnon-diffusive in origin and therefore should not depend on the diffusionconstant, and hence the size, of the molecule. The dominate mechanism ofinter-detector broadening is due to mixing and is sample-independent.Therefore if the geometrical broadening of the system can becharacterized with a reference sample, the resulting broadeningparameters can then used to correct all subsequent data runs.

Consider a simple linear model for interdetector broadening. For thepurposes of discussion, assume that the concentration detector islocated downstream of the MALS detector. Furthermore, assume that Eq.(3) is satisfied, and that data for a monodisperse reference sample hasbeen collected. Therefore, the measured concentration depends on theunderlying concentration in the MALS detector via the convolution with aparameterized broadening function B(α₀, α₁, . . . , α_(l), τ). One has

$\begin{matrix}{{{c^{m}(t)} = {\int_{- \infty}^{\infty}{{c\left( {t - \tau} \right)}{B\left( {\alpha_{0},\alpha_{1},\ldots\mspace{14mu},\alpha_{l},\tau} \right)}{\mathbb{d}\tau}}}},} & (10)\end{matrix}$

where c^(m)(t) is the measured concentration at the downstream detectorfollowing broadening, and c(t) is the concentration of the sample whenit passed through the light scattering detector. The parameters α_(l)are model specific and include the width of the broadening function andthe interdetector delay volume. The goal of the broadening correction isto find the best-fit values of these parameters. According to Eq. (5)one may write

$\begin{matrix}{{c^{m}(t)} = {\frac{1}{K^{*}M}{\int_{- \infty}^{\infty}{{R\left( {0,{t - \tau}} \right)}{B\left( {\alpha_{0},\alpha_{1},\ldots\mspace{14mu},\alpha_{l},\tau} \right)}{\mathbb{d}\tau}}}}} & (11)\end{matrix}$

Here we have used the fact that the peak is monodisperse, so that themolar mass M does not change across the peak, and it can be broughtoutside the integral. In order to determine the best fit parameters, oneforms the chi-squared difference between the downstream measuredconcentration signal, c^(m)(t), and the broadened upstream lightscattering signal and integrates it across the monodisperse sample peak

$\begin{matrix}{{\chi\left( {\alpha_{0},\ldots\mspace{14mu},\alpha_{l},\tau} \right)}^{2} = {\int_{peak}{\left( {{c^{m}(t)} - {\frac{1}{K^{*}M}{\int_{- \infty}^{\infty}{{R\left( {0,{t - \tau}} \right)}{B\left( {\alpha_{0},\ldots\mspace{14mu},\alpha_{l},\tau} \right)}{\mathbb{d}\tau}}}}} \right)^{2}{{\mathbb{d}t}.}}}} & (12)\end{matrix}$

One may then use a standard nonlinear least squares fitting package tofind the best-fit parameters for K*M and the α_(i) that minimize χ².Denote these best fit parameters α′_(i).

As a practical matter, one may include other system specific parametersin the above fit. For example, the excess Rayleigh ratio in Eq.(12) isdefined as the difference between Rayleigh ratio of the solvent withsample, and the Rayleigh ratio of the pure solvent. The Rayleigh ratioof the pure solvent can be included as an adjustable fit parameter,instead of manually determining it by setting a solvent baseline.Similarly, if the concentration detector has a slow drift, then theoffset and slope of the drift can similarly be included in the fit. Itis important to note that it is essential that the molar mass isconstant over the peak so that the inner integral in Eq.(12)

∫_(−∞)^(∞)R(0, t − τ)B(α₀, …  , α_(n), τ)𝕕τ

Once the optimal fit parameters have been determined, there are two waysin which they can be used in the subsequent analyses. One can attempt to“narrow” the concentration measurements by performing a deconvolution ofEq. (10) and attempt thereby, to answer the question: what would aconcentration detector have measured if it were coincident with thelight scattering detector? Alternatively, one can artificially broadenthe light scattering results to answer the conceptual question of whatwould the light scattering results have been if they were performeddownstream coincident with the concentration detector? We will arguethat the former method, which has formed the traditional approach, isnumerically unstable and often gives unphysical results, while thelatter method is numerically stable, but at the expense of reducing theresolution of the measurement. Since the broadening is usually a smallcorrection, the loss of resolution is minimal and the latter method ispreferable.

Consider the traditional deconvolution method. One may reconstruct c(t)from knowledge of c^(m)(t) by writing the Fourier Transform

$\begin{matrix}{{{c(t)} = {\frac{1}{\sqrt{2\pi}}{\int_{- \infty}^{\infty}{{\mathbb{e}}^{{\mathbb{i}\omega}\; t}\frac{{\overset{\sim}{c}}^{m}(\omega)}{\overset{\sim}{B}\left( {\alpha_{1}^{\prime},\ldots\mspace{14mu},\alpha_{n}^{\prime},\omega} \right)}{\mathbb{d}\omega}}}}},{where}} & (13) \\{{{{\overset{\sim}{c}}^{m}(\omega)} = {\frac{1}{\sqrt{2\pi}}{\int_{- \infty}^{\infty}{{\mathbb{e}}^{{\mathbb{i}\omega}\; t}{c^{m}(t)}{\mathbb{d}t}}}}},} & (14)\end{matrix}$

is the Fourier transform of the measured concentration, and {tilde over(B)}(α′₁, . . . , α′_(l), ω) is similarly the Fourier transform of thebroadening kernel evaluated with the fit parameters found above. One maythen compute the molar mass as a function of time by using Eq. (5). Theproblem with this procedure lies in computing the ratio {tilde over(c)}^(m)(ω)/{tilde over (B)}(α′₁, . . . , α_(l), ω) Both {tilde over(c)}^(m)(ω) and {tilde over (B)}(α′₁, . . . , α′_(l), ω) end towardszero as ω→±∞. Therefore, the ratio tends towards 0/0 which isindeterminate. Because {tilde over (c)}(ω) contains experimental noise,the ratio will have increasing large fluctuations for large ω.Therefore, when one computes c(t) using Eq. (13), the result containshigh frequency noise and unphysical “ringing,” including negativeconcentrations. While the noise can be filtered out, the unphysicalringing cannot.

The second method, which is the subject of this patent, is to broadenthe light scattering peak. Therefore, we write

$\begin{matrix}{{M(t)} = {\frac{1}{K^{*}{c^{m}(t)}}{\lim\limits_{\theta\rightarrow 0}{\int_{- \infty}^{\infty}{{R\left( {\theta,{t - \tau}} \right)}{B\left( {\alpha_{1}^{\prime},\ldots\mspace{14mu},\alpha_{l}^{\prime},\tau} \right)}\ {{\mathbb{d}\tau}.}}}}}} & (15)\end{matrix}$

This constitutes a measurement of the molar mass as a function of time,corrected for interdetector broadening. Note that right hand side of Eq.(15) contains only measurable quantities, along with the broadeningfunction evaluated with the previously determined parameters.

Models of the Broadening Function

The algorithm described above allows one to correct the light scatteringresults for the effects of interdetector band broadening given aparameterized model of the broadening kernel. However, to apply thealgorithm, one must choose a model to use. In this section we willdescribe a specific model which works well for the MALS hardware, DAWNinstruments, built by Wyatt Technology of Santa Barbara, Calif. However,it is important to note that the algorithm applies equally well to anydefinite model.

The light scattering cell used in the Wyatt MALS instrumentation is acylinder 1.25 mm in diameter fed, at 90 degrees, by a capillary that is0.1 mm in diameter, and drained by a capillary that is 0.2 mm indiameter. Typical flow rates are of the order of 1 ml/min . One candetermine whether the flow is laminar or turbulent by computing theReynolds number, defined as Re= vd/v, where v is the average flowvelocity, d is a characteristic dimension of the system, and v is thesample viscosity. For the flow in the inlet capillary one has v=127mm/sec, d=0.1 mm, and v=0.89 cm²/sec, so that Re=14. Fluid flowtypically does not become unstable until it reaches Reynolds number of afew hundred. Therefore, we conclude that the flow in the inlet capillaryis laminar. Next, consider the jet of fluid as it enters the flow cell.The flow rate is the same, but the characteristic dimension increases tod=1.25 mm. Therefore, the Reynolds number increases to Re=178.Therefore, we conclude that there is some turbulent mixing near theinlet of the cell, which has a mixing volume of some fraction of thecell volume. Similarly, there is some mixing in the inlet of aconcentration detector flow cell.

There is also some instrumental broadening which comes from the factthat the system measures a finite volume of fluid, and the measurementsystem has some intrinsic filter time constant. Therefore the model wewill consider is an exponential broadening, to represent the mixing inthe flow cell convolved with a Gaussian term to represent theinstrumental broadening. The model for the broadening kernel is

$\begin{matrix}{{{B\left( {\sigma,w,\tau} \right)} = {\frac{1}{\sigma\sqrt{2\pi}}{{\mathbb{e}}^{{{- \tau^{2}}/2}\sigma^{2}} \otimes \frac{1}{w}}{\mathbb{e}}^{{- \tau}/w}{\theta(\tau)}}},} & (16)\end{matrix}$where σ is the instrumental Gaussian width, and w is the broadening dueto mixing in the inlet of the cell and the convolution is defined as

$\begin{matrix}{{{a(t)} \otimes {b(t)}} \equiv {\int_{- \infty}^{\infty}{{a\left( {t - t^{\prime}} \right)}{b\left( t^{\prime} \right)}{{\mathbb{d}t^{\prime}}.}}}} & (17)\end{matrix}$

To test the appropriateness of this model, we performed the followingtest. We filled a 5 μl injection loop with 200 kD polystyrene dissolvedin toluene and injected it directly into the light scattering flow cell.The flow rate was 0.5 ml/min and the sample collection interval was 0.25sec. We model the data as a delta function, to represent the injectionloop, convolved with the broadening model. To compare this with the 90°light scattering signal, one forms χ² as

$\begin{matrix}{{{\chi\left( {a,t_{0},\sigma,w} \right)}^{2} = {\int_{- \infty}^{\infty}{\left\lbrack {{R\left( {90^{{^\circ}},t} \right)} - {{aB}\left( {\sigma,w,{t - t_{0}}} \right)}} \right\rbrack^{2}{\mathbb{d}t}}}},} & (18)\end{matrix}$where t₀ is the time required for the sample to flow from the injectorto the flow cell, and a is a scale factor to convert from dimensionlessunits to Rayleigh ratio. There are four fit parameters: a, t₀, σ, and w.

FIG. 2 shows the results of the fit. The light scattering data points 4are normalized so the peak has unit amplitude. The resulting fit to thebroadening function is 5. This includes only the broadening which occursbetween the injection loop and the center of the flow cell, where themeasurement takes place. It is clear the broadening function fits thedata well, and therefore is an appropriate model for interdetectorbroadening. This broadening model will be used in demonstrationsdescribed below.

Applying the Method to an Arbitrary Series of Detectors

In this section the method will be generalized for two or more arbitrarydetectors connected in series. Define the detectors responses asD_(i)(t), as the sample flows serially from D₁, to D₂, . . . to D_(n).In general, the sample peaks become progressively broader as they elutethrough the system. Any analysis on the data (D₁(t), D₂,(t), . . . ,D_(n)(t)), will be compromised by this progressive broadening. Asbefore, it is numerically more stable to broaden the upstream detectorsto be consistent with the detector with the broadest response. Usuallythis will be the last detector in the chain. However, if one of theintermediate detectors has a large instrumental broadening, the resultscan be referenced to it instead. For the purposes of discussion we willassume that the upstream detectors signals will be broadened to matchD_(n).

The procedure has two phases. In the first phase, one determines thebroadening parameters by measuring a monodisperse sample at a low enoughconcentration that the detector signals, in the absence of broadening,would be directly proportional. Practically, one injects a nearlymonodisperse sample through the chromatograph, and selects the monomersubpeak with which to determine the broadening parameters. For eachdetector i from 1 to n−1, one computes

$\begin{matrix}{{{\chi_{i}^{2}\left( {\beta_{i},\tau_{i},\alpha_{ij}} \right)} = {\int_{peak}{\left( {{D_{n}(t)} - {\beta_{i}{\int_{- \infty}^{\infty}{{D_{i}\left( {t - \tau} \right)}{B\left( {\alpha_{ij},{\tau - \tau_{i}}} \right)}{\mathbb{d}\tau}}}}} \right)^{2}{\mathbb{d}t}}}},} & (19)\end{matrix}$where χ_(i) ²(β_(i), τ_(i), α_(ij)), is the chi squared parameter forthe i^(th) detector. The parameters β_(i) are proportionality factorswhich bring the detector signals to the same scale, τ_(i) are theinter-detector delays associated with the time it takes the sample totravel from one detector to the next, and α_(ij) are the j broadeningparameters for the i^(th) detector. These χ² are minimized using astandard nonlinear least squares fit algorithm to determine the best fitparameters, β′_(i), τ′_(i) and α′_(ij). Software capable of such fittingis readily available in various commercial packages. The procedures andsome of the various software approaches have been discussed in detail byHiebert in his 1981 paper “An Evaluation of Mathematical Software ThatSolves Nonlinear Least Squares Problems” in the ACM Transactions onMathematical Software, volume 7, pages 1-16.

In the second phase, one applies the broadening parameters determined inthe first phase to correct all subsequent data runs. In this phase, allof the upstream detector signals are broadened as if they werecoincident to the last detector in the chain. Then they can be compareddirectly to the last detector D_(n). The broadened data are

$\begin{matrix}{{{D_{i}^{b}(t)} = {\int_{- \infty}^{\infty}{{D_{i}\left( {t - \tau} \right)}{B\left( {\alpha_{ij}^{\prime},{\tau - \tau_{i}^{\prime}}} \right)}{\mathbb{d}\tau}}}},} & (20)\end{matrix}$

Then the original analysis proceeds on the data sets (D₁ ^(b)(t), . . ., D_(n−1) ^(b)(t), D_(n)(t)) .

Demonstration of the Algorithm

In this section we will demonstrate the algorithm on sample data for thefollowing instrument combinations: MALS+dRI, Viscometer+dRI, andUV+MALS. FIG. 1 shows data from an 100 μl injection of BSA into a ShodexOH pack KW protein separation column manufactured by Showa Denko, Japan.The flow rate was 0.5 ml/min. The fractionated sample first passedthrough the light scattering detector and then the dRI detector. The BSAsample consisted primarily of monomers, but had a substantial content ofaggregates which were separated by the column. The peak between 16 minand 18 min primarily consists of pure monomer. The peak between 14.4 minand 16 min is pure dimer, and the subsequent peaks are higher aggregatesthat are less well separated. The horizontal axis is time in minutes.Trace 1 is the computed molar mass M_(w)(t) divided by the monomer molarmass M_(w0). Therefore the monomer reads 1 on the vertical axis; thedimer reads 2, etc. Trace 2 is the 90° light scattering signal and trace3 is the refractive index signal. Both the refractive index and thelight scattering have been normalized so that their peaks have the samemaximum height. The vertical scale for traces 2 and 3 is arbitrary.

The first step in the algorithm is to choose the broadening model. Forthe data set in FIG. 1, the hybrid model described in Eq.(16) was used.The second step in the algorithm is to choose a monodisperse peak onwhich to determine the fit parameters. The monomer peak between 16 mmand 18 mm was used. When one forms the χ² in Eq. (19), there are fourparameters to be determined by nonlinear least squares fitting. An extraparameter was added to account for a possible baseline shift between thetwo data sets. Therefore the χ² that was minimized was

$\begin{matrix}{{{\chi^{2}\left( {\beta,\tau_{0},\sigma,w,x_{0}} \right)} = {\int_{peak}{\left( {{\mathbb{d}{{RI}(t)}} - {\beta{\int_{- \infty}^{\infty}{{{LS}\left( {t - \tau} \right)}{B\left( {\sigma,w,{\tau - \tau_{0}}} \right)}{\mathbb{d}\tau}}}} + x_{0}} \right)^{2}{\mathbb{d}t}}}},} & (21)\end{matrix}$where, LS (t) is the 90° light scattering signal as a function of time,and dRI (t), is the dRI data as a function of time. This model wasminimized by using a commercial Marquardt nonlinear least squarespackage such as described by D. W. Marquardt in his 1963 article in theJournal of the Society of Industrial and Applied Mathematics, “Analgorithm for least squares estimation of nonlinear parameters,” volume11, pages 431 to 441. Note that to perform the broadening correctionproscribed by Eq. (20), one does not need β, or x₀, so they will not bepresented in the results table below, but they were included to insurethat the nonlinear minimization worked correctly.

Parameter Value σ′ 0.0636 sec w′  3.178 sec τ′₀    21 sec

These results can be interpreted by noting that <<w, implying that, asexpected, the dominant broadening effect is mixing, not Gaussiandiffusion or instrumental broadening. The τ₀ parameter simply indicatesthat the sample took 21 seconds to traverse between the two instruments.In this context Eq. (20) can be rewritten as

$\begin{matrix}{{{LS}^{b}(t)} = {\int_{- \infty}^{\infty}{{{LS}\left( {t - \tau} \right)}{B\left( {\sigma^{\prime},w^{\prime},{\tau - \tau_{0}^{\prime}}} \right)}{{\mathbb{d}\tau}.}}}} & (22)\end{matrix}$

FIG. 3 shows the broadened light scattering signal 7 computed using Eq.(22) with the fit parameters found in the first step above. The lightscattering analysis proceeds as before using Trace 7 instead of theoriginal light scattering data. The results are shown in Trace 6. Thereare a number of features to notice. First is that Trace 7 now overlapsTrace 2 at the monomer peak. This shows that model is correct, and thatthe nonlinear minimization worked properly. The second observation isthat the dimer peak, between 14.5 mm and 16 mm, is now also a plateau asexpected. The trimer peak, between 13.5 mm and 14.5 mm is also flatter,but since it was not baseline separated from its neighboring peaks, theresults are less clear. The next observation is that the plateaus arenow integral multiples of each other. This is to be expected since themolar mass of the dimer should be exactly twice the mass of the monomer.Similarly, the trimer plateau is now three times that of the monomer.

FIG. 4 shows data from an 100 μl injection of BSA into a Shodex OH packKW protein separation column. The flow rate was 0.5 ml/min. Thefractionated sample first passed an online bridge viscometer and thenthrough a dRI detector. The viscometer measures the specific viscositywhich is defined asη_(sp)(t)=η(t)/η₀−1,   (23)where η(t) is the viscosity of the fluid sample combination and η₀ isthe viscosity of the pure fluid. The dRI detector measures therefractive index of the sample which can be converted to concentrationif the sample dn/dc is known. One can then compute the intrinsicviscosity defined as

$\begin{matrix}{{\eta_{int}(t)} = {\lim\limits_{c\rightarrow 0}{{\eta_{sp}(t)}/{{c(t)}.}}}} & (24)\end{matrix}$

If the concentration is sufficiently low then the intrinsic viscositycan be approximated asη_(int)(t)≈η_(sp)(t)/c(t).   (25)

Trace 9 is the specific viscosity, and trace 10 is the dRI signal.Traces 9 and 10 have been normalized to have the same maximum height.The vertical scale for these two traces is arbitrary. The results ofthis measurement are shown in Trace 8 which presents the intrinsicviscosity on the vertical axis in units of ml/g. The intrinsic viscosityshould be constant across each sample peak. The curvature seen is due tointerdetector broadening, but in this case the viscometer signal issubstantially broader than the dRI signal. This is because theviscometer measures the viscosity over a much larger fluid volume thanthe dRI. In this example, the dominant broadening is instrumentalbroadening internal to the viscometer. In fact, the dRI signal isbroadened by the passage of the sample between the two detectors, whichreduces the overall difference in peak widths. The method of thisinvention can be applied as before, but in this case, the dRI signalwill be broadened to match the viscosity signal. The first step in thealgorithm is to choose a broadening model. Again the hybridGaussian-mixing model was used. The second step in the algorithm is tochoose a monodisperse peak on which to determine the fit parameters. Themonomer peak between 15.2 min and 16.2 min was used. As before Eq. (19)can be rewritten as

$\begin{matrix}{{{\chi^{2}\left( {\beta,\tau_{0},\sigma,w,x_{0}} \right)} = {\int_{peak}{\left( {{\eta_{sp}(t)} - {\beta{\int_{- \infty}^{\infty}{{\mathbb{d}{{RI}\left( {t - \tau} \right)}}{B\left( {\sigma,w,{\tau - \tau_{0}}} \right)}{\mathbb{d}\tau}}}} + x_{0}} \right)^{2}{\mathbb{d}t}}}},} & (26)\end{matrix}$and the fit parameters are determined by minimizing χ² with respect tothe adjustable parameters. We find the fit parameters

Parameter Value σ′ 0.01 sec w′ 9.04 sec τ′₀ 15.6 sec

In this case, the parameter w is 9.04 seconds, which arises not so muchfrom inter-detector mixing, but from the large internal volume of theviscometer capillaries relative to the volume of the dRI flow cell. TheGaussian term is extremely small, which again indicates the diffusion isinsignificant. The dRI signal may be broadened by rewriting Eq. (20) as

$\begin{matrix}{{{\mathbb{d}{{RI}^{b}(t)}} = {\int_{- \infty}^{\infty}{{\mathbb{d}{{RI}\left( {t - \tau} \right)}}{B\left( {\sigma^{\prime},w^{\prime},{\tau - \tau_{0}^{\prime}}} \right)}{\mathbb{d}\tau}}}},} & (27)\end{matrix}$and the analysis can proceed with the broadened dRI signal. FIG. 5 showsthe broadened dRI signal in Trace 12 and the corrected intrinsicviscosity signal is shown in Trace 11. The results now consist ofplateaus for each peak.

The last example, illustrated in FIG. 6, consists of a 100 μl sample of3.0 mg/ml BSA measured by a UV detector followed by a MALS detector. Inthis case, the LS signal shown at 14 is significantly broader than theUV signal at 15, giving rise to the curved molar mass trace shown at 13.For this example, the UV signal will be broadened to match the LSsignal. Equation (19) is rewritten as

$\begin{matrix}{{\chi^{2}\left( {\beta,\tau_{0},\sigma,w,x_{0}} \right)} = {\int_{peak}{\left( {{{LS}(t)} - {\beta{\int_{- \infty}^{\infty}{{{UV}\left( {t - \tau} \right)}{B\left( {\sigma,w,{\tau - \tau_{0}}} \right)}{\mathbb{d}\tau}}}} + x_{0}} \right)^{2}{{\mathbb{d}t}.}}}} & (28)\end{matrix}$

Again the hybrid mixing and Gaussian model is used, and the monomer peakbetween 13.5 min and 15 min is used to determine the fit parameters.Minimizing χ² with respect to the fit parameters yields

Parameter Value σ′ 0.22 sec w′ 3.61 sec τ′₀ 39.5 sec

Again, the dominant source of broadening is internal mixing. The UV datais now broadened, as shown in FIG. 7 by computing Eq. (20) rewritten as

$\begin{matrix}{{{UV}^{b}(t)} = {\int_{- \infty}^{\infty}{{{UV}\left( {t - \tau} \right)}{B\left( {\sigma^{\prime},w^{\prime},{\tau - \tau_{0}^{\prime}}} \right)}{{\mathbb{d}\tau}.}}}} & (29)\end{matrix}$

The broadened UV signal is shown in Trace 17, and the molar mass isshown in Trace 16. As with the previous two cases, the data consists ofplateaus, and the effects of inter-detector broadening are dramaticallyreduced.

The foregoing demonstration of the inventive method illustrates but afew of the possible types of band broadening corrections that may beaddressed and corrected by the inventive methods disclosed in thisspecification. As will be obvious to those skilled in thechromatographic arts, until this invention, the effects of bandbroadening have represented significant impediments to improving theprecision of measurements associated with the use of multiple detectors.This inventive method and all its variants fulfill the expectation thatband broadening effects will thereby be reduced dramatically.Accordingly, I claim

The invention is clearly described above in a manner that isunderstandable to one skilled in the mathematical data analysis arts;however, in order to make the invention more readily understandable,FIG. 8 is presented which graphically illustrates the method ofcorrecting for band broadening using Eq. (19). FIG. 8 a shows detectorsignals, D₁ (t), 18, and D₂ (t), 19. The function presented in Eq. (19)can be decomposed into the following steps. First the detector signalwith the narrowest peak, 19, is identified as the signal that will becorrected, i. e. will be broadened. Signal 19 is then shifted by theinterdetector time, τ₂ so it comes into alignment with signal 18. Asshown in FIG. 8 b, it is scaled by β₂ so that it is on the samescale/height as the broadest peak, 18. This newly processed signal peak20 is then broadened with the model B(α_(2j),τ−τ₂). As shown in FIG. 8c, the broadened data 21 is finally compared to the data of theunprocessed broad peak 18 by computing a χ² model which is thenminimized to determine optimal values of β₂,α_(2j),τ₂. FIG. 8 d comparesthe data after the χ² minimization has been completed, i. e., the peaksfrom the two detectors overlay. It should be stressed that Eq. (19)accomplishes all of these steps simultaneously, as would be obvious tothose trained in the art. Although FIG. 8 has been shown in sections forpedagogical reasons, the execution of the least squares process of Eq.(19) is done in a single calculation.

1. A method to determine the best fit parameters of a broadening modelto be used to correct for the effects of interdetector band broadeningin a chromatographic separation containing a separation device followedby two or more detectors i (=1,n), where n is the number of detectors,comprising the steps of A. Selecting a broadening model B(α₀, α₁, . . ., α_(l), t) containing a set of adjustable parameters α_(j)(=0, . . . ,l), where l is the number of adjustable parameters; B. Injecting areference sample of uniform composition; C. Collecting the i detectorsignals D_(i)(t) as a function of time corresponding to a peak of saidsample from each of said detectors of where a peak is defined as a rangeof time during which the sample of uniform composition elutes; D.Defining the reference signal D_(n)(t) as that which exhibits thebroadest temporal response; E. For each detector i, forming a χ_(i) ²model which is derived from said broadening model, and is to beminimized over said peak using said signal D_(n)(t) as a referenceagainst which the said detector i is to be compared;χ_(i)²(β_(i), τ_(i), α_(ij)) = ∫_(peak)(D_(n)(t) − β_(i)∫_(−∞)^(∞)D_(i)(t − τ)B(α_(ij), τ − τ_(i))𝕕τ)²𝕕t,where said best fit parameters are β_(i), α_(ij), and τ_(i); and (a)β_(i) is the scale factor for detector i; (b) α_(ij) characterizes theextent of the broadening for detector i and parameter j; and (c) τ_(i)is the interdetector time delay for detector i. F. Minimizing said χ_(i)² models to determine yield said best fit parameters for said detectorpeak.
 2. The method of claim 1 where the minimization of said χ_(i) ²models are achieved by use of a nonlinear least squares algorithm. 3.The method of claim 2 where said nonlinear least squares algorithm is ofthe type developed by Marquardt.
 4. The method of claim 1 where saidinterdetector band broadening is caused by dilution.
 5. The method ofclaim 1 where said interdetector band broadening is caused by mixing. 6.The method of claim 1 where said interdetector band broadening is causedby mechanical defects within the detector cells andlor connectorsthereto.
 7. The method of claim 1 where said interdetector bandbroadening is caused by internal instrumental averaging.
 8. The methodof claim 7 where said internal instrumental averaging is from electronicfiltering.
 9. The method of claim 7 where said internal instrumentalaveraging is by measuring a range of volumes of the sample.
 10. Themethod of claim 1 where said peaks of uniform composition correspond tomonodisperse fractions which are separated from the other fractions bysaid chromatographic separation.
 11. The method of claim 1 where saidpeaks of uniform composition correspond respectively to fractions forwhich said separation device is ineffective and produces no appreciableseparation so that said sample elutes with a uniform composition. 12.The method of claim 1 where said broadening model is given by${{B\left( {\alpha_{1},\alpha_{2},t} \right)} = {\int_{- \infty}^{\infty}{\frac{1}{\alpha_{1}\sqrt{2\pi}}{\mathbb{e}}^{{{- \tau^{2}}/2}\alpha_{1}^{2}}\frac{1}{\alpha_{2}}{U\left( {t - \tau} \right)}{\mathbb{e}}^{{- {({t - \tau})}}/\alpha_{2}}{\mathbb{d}\tau}}}},$where α₁ and α₂ are parameters that characterize said interdetector bandbroadening; the parameter α₁ characterizes Gaussian broadening due tointernal instrumental averaging, and α₂ characterizes an interdetectormixing volume; and where U(t−τ)=1 when t≧τ and U(t−τ)=0 when t<τ.